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Homework Statement:

An artificial satellite circles the Earth in a circular orbit
at a location where the acceleration due to gravity is
9.00 m/s2. Determine the orbital period of the satellite.
Relevant Equations:

a = V^2/r
T = 2 Pi R / V
F = G M m / R^2
T^2 = 4 Pi^2 a^3 / (G M)  Kepler's third law
I didn't use Kepler's 3rd law and this may be the reason I have a wrong answer.
However, I want to know: where I make the mistake.
## ma = \frac {G M m} {R^2}##
## R = (\frac {G M} a)^{1/2}##
##a = \frac {v^2} R##
##V^2 = a R = a (\frac {G M} a)^{1/2} = (G a M)^{1/2}##
##V = (G a M)^{1/4}##
##T = \frac {2 \Pi R} V##
##T = \frac {2 \Pi (G M)^{1/2}} {(G M a)^{1/4} a^{1/2}} = \frac {2 \Pi(G M)^{1/4}} {a^{3/4}}##
However, I want to know: where I make the mistake.
## ma = \frac {G M m} {R^2}##
## R = (\frac {G M} a)^{1/2}##
##a = \frac {v^2} R##
##V^2 = a R = a (\frac {G M} a)^{1/2} = (G a M)^{1/2}##
##V = (G a M)^{1/4}##
##T = \frac {2 \Pi R} V##
##T = \frac {2 \Pi (G M)^{1/2}} {(G M a)^{1/4} a^{1/2}} = \frac {2 \Pi(G M)^{1/4}} {a^{3/4}}##